ON THE MOTION OF PENDULUMS. 117 lation, and in fact T~l loge (a0O differs veiT sensibly from A' Making now this substitution in (174), integrating, and after integration restoring a in the last term by means of (175), we get g log a2 - log a0 0 being an arbitrary constant. To determine the three constants A, B} 0, let Oj be the arc observed at the middle of the experiment, apply the last equation to the arcs a0, alt «2, and take the first and second differences of each member of the equation. Let At denote the sum of the two first differences, so that A^ is the same thing as T. Then we may take for the two equations to determine A and B '•*•*; A2log*0 = -^ Eliminating B, and passing from Napierian to common logarithms, which will be denoted by Log., we get — A Loof ot ~ Log e. Ajtf _ .(177). If we suppose the part of — -j- which does not vary as the first cLt power of a to be a2<£' (a) instead of jBa2, we shall get in the same way ......( }' 76. I have not attempted to deduce evidence for or against the truth of equation (173) from Bessel's experiments. The approximate formula (175) so nearly satisfied the observations, that almost any reasonable formula of interpolation which introduced one new disposable constant would represent the experiments within the limits of errors of observation. It may be observed, that the factor outside the brackets in equations (177) and (178) is the first approximate value of A got by using only the initial and final arcs, and supposing the arcs to decrease iu geometric progression. In the case of the long pendulum, the value of A, corrected in accordance with the formula (178), would be very sensibly different according as we supposed (f> (a) to be equal to Ba, in which case (178) would reduce itself to (177), or equal to I: I lit i